What is the integral of sin^{10}(x)cos^3(x) ?

The integral of sin^{10}(x)cos^3(x) is (sin^{11}(x))/(11)-(sin^{13}(x))/(13)+C

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integral of sin^{10}(x)cos^3(x)

\int sin^{10} (x) cos^{3} (x) d x

\frac{sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13} + C

Solution steps

\int sin^{10} (x) cos^{3} (x) d x

Rewrite using trig identities

= \int (1 - sin^{2} (x)) cos (x) sin^{10} (x) d x

Apply u-substitution

: \int u^{10} (1 - u^{2}) d u

= \int u^{10} (1 - u^{2}) d u

Expand

u^{10} (1 - u^{2}) : u^{10} - u^{12}

= \int u^{10} - u^{12} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{10} d u - \int u^{12} d u

\int u^{10} d u = \frac{u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= \frac{u ^{11}}{11} - \frac{u ^{13}}{13}

Substitute back

u = sin (x)

= \frac{sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13}

Add a constant to the solution

= \frac{sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13} + C

\int sin^{3} (2 x) cos^{2} (2 x) d x

x \to 1 lim (\frac{( sin ( x ) )}{x})

y^{^{'}} sin (x) = y ln (y)

f (x) = \frac{x ^{2} - 1}{( x + 1 ) ^{2}}

ln (1 + 4 x)

What is the integral of sin^{10}(x)cos^3(x) ?
The integral of sin^{10}(x)cos^3(x) is (sin^{11}(x))/(11)-(sin^{13}(x))/(13)+C